Inverse designed Hamiltonians for perfect state transfer and remote entanglement generation, and applications in superconducting qubits

TL;DR

This paper develops an inverse-design framework for robust quantum state transfer by constructing the dome model, a tridiagonal, mirror-symmetric Hamiltonian whose eigen-spectrum is engineered via the inverse eigenvalue method. A tunable parameter $m$ shapes the energy gaps, enabling simultaneous perfect state transfer and remote entanglement generation within a single evolution, while large $m$ induces effective edge-only SWAP dynamics that resist intermediate-qubit noise. The authors analyze the dynamics, noise resilience, and scalability in 1D and 2D qubit networks, and propose a cascaded PST/FST scheme to achieve long-distance transfer with favorable time-performance tradeoffs. They also discuss extensions to other architectures and highlight directions for relaxing design constraints and exploring broader eigenvalue spectra. Overall, the dome model offers a practical, analytically tractable route to robust, scalable quantum interconnects for superconducting qubits and related platforms.

Abstract

Hamiltonian inverse engineering enables the design of protocols for specific quantum evolutions or target state preparation. Perfect state transfer (PST) and remote entanglement generation are notable examples, as they serve as key primitives in quantum information processing. However, Hamiltonians obtained through conventional methods often lack robustness against noise. Assisted by inverse engineering, we begin with a noise-resilient energy spectrum and construct a class of Hamiltonians, referred to as the dome model, that significantly improves the system's robustness against noise, as confirmed by numerical simulations. This model introduces a tunable parameter $m$ that modifies the energy-level spacing and gives rise to a well-structured Hamiltonian. It reduces to the conventional PST model at $m=0$ and simplifies to a SWAP model involving only two end qubits in the large-$m$ regime. To address the challenge of scalability, we propose a cascaded strategy that divides long-distance PST into multiple consecutive PST steps. Our work is particularly suited for demonstration on superconducting qubits with tunable couplers, which enable rapid and flexible Hamiltonian engineering, thereby advancing the experimental potential of robust and scalable quantum information processing.

Figures (9)

• Figure 1: Schematic of perfect state transfer (PST) and fractional state transfer (FST). (a) In PST, a quantum state initialized at $\ket{1}$ (or more generally, $\ket{n}$) is perfectly transferred to $\ket{N}$ ($\ket{N-n+1}$) after a transfer time $\tau_{\text{PST}}$. (b) In FST, a quantum state initialized at $\ket{1}$ ($\ket{n}$) evolves into a superposition of $\ket{1}$ and $\ket{N}$ ($\ket{n}$ and $\ket{N-n+1}$) after a certain time $\tau_{\text{FST}}$. The amplitudes and relative phase between $\ket{1}$ and $\ket{N}$ ($\ket{n}$ and $\ket{N-n+1}$) can be controlled by engineering the system Hamiltonian.
• Figure 2: Eigenenergies, eigenfunctions, and Hamiltonian parameters of the line and dome model. (a)--(b) Frequency and coupling configurations of the line and dome model, respectively. The vertical axis represents qubit frequencies, and the thickness of the gray lines indicates the relative coupling strength between adjacent qubits. (c)--(d) Eigenenergies and corresponding eigenfunctions of the line and dome model, respectively. In the line model, all eigenenergies are equally spaced. In the dome model, all energy levels are pushed upward with increasing $m$, except for the two lowest levels, which remain unchanged. (e)--(f) Hamiltonian parameters, including qubit frequencies $\omega_n$ and coupling strengths $J_n$, for various values of $m$. Parameters with $m=0$ correspond to the line model, while those with $m>0$ correspond to the dome model.
• Figure 3: Numerical results of (a) the effective frequencies $\omega_{1, N}^\text{eff}$ and (b) the effective coupling strength $J^\text{eff}$ between the two end qubits in the large-$m$ limit. The calculation is based on a perturbative Schrieffer-Wolff transformation method, which block-diagonalizes the full $N \times N$ single-excitation Hamiltonian into a $2\times2$ subspace involving sites 1 and $N$, and an $(N-2)\times(N-2)$ subspace involving the remaining sites. This approximation holds only when $m$ is sufficiently large; for small $m$, the perturbative treatment breaks down, leading to divergent numerical results, which are not shown in the figure.
• Figure 4: Qubit dynamics under different $m$ values, simulated on a $1\times5$ qubit chain and its reduction from PST to SWAP. (a)--(c) Time evolution of qubit populations for $m=0$, 2 and 102, respectively. For $m\ne0$, both FST and PST occur sequentially at $t/T=0.25$ and 0.5, while for $m=0$, only PST is observed at $t/T=0.5$. In case (c), when $m \gg 1$, the PST dynamics reduces to the SWAP dynamics between the two end sites, with effective coupling given by $J^\text{eff} = \pm J/2$; see Eq. \ref{['eq-JefflargeN']}. (d)--(f) Density matrices of the two endpoint qubits $Q_1$ and $Q_5$ at $t/T=0$, 0.25 and 0.5, respectively. The bar height represents the amplitude of each element in the density matrix, while the color encodes its phase. The simulations are performed under ideal and noise-free conditions, the density matrix in (d) corresponds to $t=0$ for all $m$ values; (e) corresponds to the result indicated by the light yellow dashed line in (b) and (c); and (f) corresponds to the result indicated by the dark yellow dashed line in (a)--(c).
• Figure 5: Impact of different types of coherent noise on the fidelities of remote entanglement generation and quantum state transfer, simulated on a $1\times5$ qubit chain. (a)--(d) Fidelities of entanglement generation at $t=0.25T$ under various noise types. (a) Noise on the middle qubit frequencies $\omega_{\text{middles}}$ ($Q_2$--$Q_4$); (b) Noise on the edge qubit frequencies $\omega_{\text{edges}}$ ($Q_1$ and $Q_5$); (c) Noise on the coupling strengths $J_n$ ($J_1$--$J_4$); (d) Noise on all components. Noise is sampled from a Gaussian distribution centered at the ideal parameter values with a variance of $\sigma^2$. Each data point represents the average fidelity over 100 samples, with the error bars indicating the standard deviation. The fidelity of entanglement generation is defined as the overlap between the density matrix $\rho$ of the two edge qubits at $t=0.25T$ and the ideal density matrix $\dyad{\psi}{\psi}$, i.e., $F(t=0.25T) = \mathrm{Tr}(\rho \dyad{\psi}{\psi})$, where $\ket{\psi} = (\ket{\downarrow\uparrow} + i\ket{\uparrow\downarrow})/\sqrt{2}$ is a Bell entangled state. (e)--(h) Fidelities of quantum state transfer at $t=0.5T$ under the same noise types as (a)--(d), respectively. Each data point is averaged over 50 samples. Fidelity is defined as the overlap between the reconstructed process matrix $\chi$, obtained via the quantum process tomography simulation, and the ideal matrix $\chi_{\text{ideal}}$, i.e., $F(t=0.5T) = \mathrm{Tr}(\chi \chi_{\text{ideal}})$. Details of the reconstruction of $\chi_{\text{ideal}}$ are provided in the main text.